In Philosopher's Stone, Dumbledore suspected that Quirrell is up to something, and ordered Snape to keep an eye on Quirrell. In Goblet of Fire, Moody is the Defense Against the Dark Arts teacher, but he is just a disguised Barty Crouch Jr. No one in Hogwarts suspects Moody as an impostor, even Dumbledore can't sense it. Why didn't Snape suspect him? Like keeping an eye on him, like he did to Quirrell.

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Are you asking why Snape didn't suspect Moody, or why Rowling chose for him not to? "Couldn't" implies Snape was incapable of suspecting Moody, in reality any character could have suspected anyone of anything, they just mostly didn't because there was no reason for them to.
– DaaaahWhooshJun 21 '17 at 18:05

1 Answer
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Because Barty Crouch Jr was a very good impostor

Quirrell was a disturbed man behaving quite oddly. He was known to be in Albania, a place closely linked to Voldemort. Dumbledore had every reason to ask Snape to pay special attention to that guy.

Moody in Goblet Of Fire is quite different. Nobody suspects him until he is revealed by the end of the book. He made a good act, very convincing. Snape could suspect that an impostor is in the Castle because ingredients that could be used for Polyjuice Potion were missing. However, he naturally suspected Harry and his friends (who did this in the past, whether Snape knew that or not). To suspect Moody would be a very crazy idea at the time. Also, Snape's reputation was worse than that of Moody due to his past misgivings, so if he voiced his suspicions, nobody would treat them seriously.

Out-of-universe remark:

Even the reader (or watcher of the movie) does not suspect Moody a tiny bit.

I disagree on the movie bit - they totally telegraph that Moody is Crouch entirely too early with that stupid tongue flicking thing. I generally like Tennant but his choices (or direction) were way over the top
– NKCampbellJun 21 '17 at 18:08

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@NKCampbell: Completely agree, the fake Moody tongue flick and the Crouch tongue flick should have been further apart (movie time wise) to make it less hilariously obvious.
– FlaterJun 22 '17 at 8:07